On a smooth quartic surface containing 56 lines which is isomorphic as a K3 surface to the Fermat quartic

TL;DR

This paper explicitly defines a complex smooth quartic surface with 56 lines, explores its reductions in positive characteristic, and establishes its isomorphism to the Fermat quartic surface, which has only 48 lines.

Contribution

It provides an explicit defining equation for a quartic surface with 56 lines and demonstrates its isomorphism to the Fermat quartic, revealing new geometric properties.

Findings

The surface contains 56 lines over complex numbers.

The surface is isomorphic to the Fermat quartic surface.

Reductions to positive characteristic are investigated.

Abstract

We give a defining equation of a complex smooth quartic surface containing 56 lines, and investigate its reductions to positive characteristics. This surface is isomorphic to the complex Fermat quartic surface, which contains only 48 lines. We give the isomorphism explicitly.